A201998 Positive numbers n such that n^2 + n + 41 is composite and there are no positive integers c such that n = c*x^2 + (c + 1)*x + c*41 for an integer x.

244, 249, 251, 266, 270, 295, 301, 336, 344, 389, 399, 407, 416, 418, 445, 449, 454, 466, 489, 494, 496, 500, 506, 527, 531, 545, 547, 563, 570, 571, 582, 583, 585, 611, 612, 620, 622, 624, 628, 630, 636, 652, 661, 662, 663, 679, 693, 699

Offset

1,1

Comments

The composition of functions k(x) factors. k(x) = (x^2 + x + 41)*(c^2*x^2 + (c^2 + 2*c)*x + c^2*41 + c + 1). So k(x) is the product of two integers greater than one and thus composite.

References

John Stillwell, Elements of Number Theory, Springer, 2003, page 3.

Links

Table of n, a(n) for n=1..48.

Matt C. Anderson A prime producing polynomial writeup

Maple

maxn:=1000:
A:={}:
for n from 1 to maxn do
g:=n^2+n+41:
if isprime(g)=false then
A:=A union {n}:
end if:
end do:
# The set A contains values n such that n^2+n+41 is composite and n < maxn.
c:=1:
x:=-1:
p:=41:
q:=c*x^2-(c+1)*x+c*p:
A2:=A:
while q < maxn do
while q < maxn do
A2:=A2 minus {q}:
A2:=A2 minus {c*x^2+(c+1)*x+c*p}:
x:=x+1:
q:=c*x^2-(c+1)*x+c*p:
end do:
c:=c+1:
x:=-1:
q:=c*x^2-(c+1)*x+c*p:
end do:
A2;

Mathematica

Reap[For[n=1, n<700, n++, If[!PrimeQ[n^2+n+41], If[Reduce[c>0 && n == c*x^2+(c+1)*x+41*c , {c, x}, Integers] === False, Sow[n]]]]][[2, 1]] (* Jean-François Alcover, Apr 30 2014 *)

Crossrefs

Cf. A007634 (n^2 + n + 41 is composite).

Cf. A235381 (similar to this sequence).

Sequence in context: A243774 A051002 A044987 * A234262 A031786 A328206

Adjacent sequences: A201995 A201996 A201997 * A201999 A202000 A202001

Keyword

nonn

Author

Matt C. Anderson, Dec 07 2011

Status

approved